Ruang Contoh dan Peluang Pertemuan 05 Matakuliah : L0104 / Statistika Psikologi Tahun : 2008 Ruang Contoh dan Peluang Pertemuan 05

Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Mahasiswa akan dapat menghitung dan memahami peluang kejadian tunggal dan majemuk. 3 Bina Nusantara

Outline Materi Ruang sampel, kejadian dan peluang kejadian Operasi gabungan dan irisan antar himpunan (Intersection) Kaidah komplemen dan Saling meniadakan (mutually exclusive) Kaidah penjumlahan/gabungan peluang (Union) 4 Bina Nusantara

What is Probability? Probabilitas Dalam Statistika kita dihadapkan untuk menarik kesimpulan dan keputusan dari suatu permasalahan. Kesimpulan yang dibuat, kebenarannya tidaklah pasti secara absolut, sehingga timbul persoalan bagaimana keyakinan untuk mempercayai ke-benaran dari kesimpulan tersebut. Untuk hal tersebut diperlukan suatu teori yang biasa disebut teori peluang atau probabilitas. Dalam teori ini dibahas, antara lain tentang ketidak pastian dari suatu kejadian atau peristiwa. Probabilitas ialah suatu nilai yang digunakan untuk mengukur tingkat terjadinya suatu kejadian yang acak Bina Nusantara

Relative frequency = f/n What is Probability? We measured “how often” using Relative frequency = f/n As n gets larger, Sample And “How often” = Relative frequency Population Probability Bina Nusantara

Ruang Sampel dan Kejadian Ruang sampel merupakan kumpulan dari semua kejadian/peristiwa yang mungkin terjadi dari suatu percobaan. Dengan kata lain ruang sampel adalah himpunan semesta dari semua titik sampel dari suatu percobaan. Misalkan H = menyatakan kejadian “Head” dan T = menyatakan kejadian “Tail” pada pelemparan mata uang, maka S = { H, T } menyatakan himpunan kejadian “head” dan “tail” yang disebut himpunan semesta (S) atau ruang himpunan kejadian, atau disebut sebagai ruang sampel. Sedangkan H dan T disebut titik sampel. . Bina Nusantara

Basic Concepts An experiment is the process by which an observation (or measurement) is obtained. An event is an outcome of an experiment, usually denoted by a capital letter. The basic element to which probability is applied When an experiment is performed, a particular event either happens, or it doesn’t! Bina Nusantara

Experiments and Events Experiment: Record an age A: person is 30 years old B: person is older than 65 Experiment: Toss a die A: observe an odd number B: observe a number greater than 2 Bina Nusantara

Not Mutually Exclusive Basic Concepts Two events are mutually exclusive if, when one event occurs, the other cannot, and vice versa. Experiment: Toss a die A: observe an odd number B: observe a number greater than 2 C: observe a 6 D: observe a 3 Not Mutually Exclusive B and C? B and D? Mutually Exclusive Bina Nusantara

Basic Concepts An event that cannot be decomposed is called a simple event. Denoted by E with a subscript. Each simple event will be assigned a probability, measuring “how often” it occurs. The set of all simple events of an experiment is called the sample space, S. Bina Nusantara

Example S ={E1, E2, E3, E4, E5, E6} E1 E2 E3 E4 S E5 E1 E3 E6 E5 E2 E6 The die toss: Simple events: Sample space: 1 E1 E2 E3 E4 E5 E6 S ={E1, E2, E3, E4, E5, E6} 2 S E1 E6 E2 E3 E4 E5 3 4 5 6 Bina Nusantara

The die toss: Basic Concepts A ={E1, E3, E5} B ={E3, E4, E5, E6} S A B An event is a collection of one or more simple events. S A B E1 E3 The die toss: A: an odd number B: a number > 2 E5 E2 E6 E4 A ={E1, E3, E5} B ={E3, E4, E5, E6} Bina Nusantara

The Probability of an Event The probability of an event A measures “how often” we think A will occur. We write P(A). Suppose that an experiment is performed n times. The relative frequency for an event A is If we let n get infinitely large, The relative frequency of event A in the population Bina Nusantara

The Probability of an Event P(A) must be between 0 and 1. If event A can never occur, P(A) = 0. If event A always occurs when the experiment is performed, P(A) =1. The sum of the probabilities for all simple events in S equals 1. The probability of an event A is found by adding the probabilities of all the simple events contained in A. Bina Nusantara

Example Toss a fair coin twice. What is the probability of observing at least one head? 1st Coin 2nd Coin Ei P(Ei) H HH 1/4 P(at least 1 head) = P(E1) + P(E2) + P(E3) = 1/4 + 1/4 + 1/4 = 3/4 H T HT H TH T T TT Bina Nusantara

You can use counting rules to find nA and N. If the simple events in an experiment are equally likely, you can calculate You can use counting rules to find nA and N. Bina Nusantara

Permutations The number of ways you can arrange n distinct objects, taking them r at a time is Example: How many 3-digit lock combinations can we make from the numbers 1, 2, 3, and 4? The order of the choice is important! Bina Nusantara

Combinations The number of distinct combinations of n distinct objects that can be formed, taking them r at a time is Example: Three members of a 5-person committee must be chosen to form a subcommittee. How many different subcommittees could be formed? The order of the choice is not important! Bina Nusantara

Event Relations The union of two events, A and B, is the event that either A or B or both occur when the experiment is performed. We write A B S A B Bina Nusantara

Event Relations The complement of an event A consists of all outcomes of the experiment that do not result in event A. We write AC. S AC A Bina Nusantara

Calculating Probabilities for Unions and Complements There are special rules that will allow you to calculate probabilities for composite events. The Additive Rule for Unions: For any two events, A and B, the probability of their union, P(A B), is A B Bina Nusantara

Calculating Probabilities for Complements AC Calculating Probabilities for Complements We know that for any event A: P(A AC) = 0 Since either A or AC must occur, P(A AC) =1 so that P(A AC) = P(A)+ P(AC) = 1 P(AC) = 1 – P(A) Bina Nusantara

Selamat Belajar Semoga Sukses. Bina Nusantara